compute ( f(c + h)-f(c) ) at the indicated point.\n( f(x)=6 x^{2} ; c = 3 )\n( f(c + h)-f(c)=quad ) at ( c =…

compute ( f(c + h)-f(c) ) at the indicated point.\n( f(x)=6 x^{2} ; c = 3 )\n( f(c + h)-f(c)=quad ) at ( c = 3 ).\n(type an expression using ( h ) as the variable.)

compute ( f(c + h)-f(c) ) at the indicated point.\n( f(x)=6 x^{2} ; c = 3 )\n( f(c + h)-f(c)=quad ) at ( c = 3 ).\n(type an expression using ( h ) as the variable.)

Answer

Explanation:

Step1: Find ( f(c + h) )

Given ( f(x)=6x^{2}) and ( c = 3), then ( f(c + h)=f(3 + h)=6(3 + h)^{2}). Using the formula ((a + b)^{2}=a^{2}+2ab + b^{2}) with (a = 3) and (b=h), we have (f(3 + h)=6(9 + 6h+h^{2})=54+36h + 6h^{2}).

Step2: Find ( f(c) )

Since (c = 3), then (f(c)=f(3)=6\times3^{2}=6\times9 = 54).

Step3: Compute ( f(c + h)-f(c) )

Substitute the expressions for (f(c + h)) and (f(c)) into (f(c + h)-f(c)): [ \begin{align*} f(3 + h)-f(3)&=(54+36h + 6h^{2})-54\ &=36h+6h^{2} \end{align*} ]

Answer:

(6h^{2}+36h)